Rice formula for processes with jumps and applications
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We extend Rice Formula to a process which is the sum of two independent processes: a smooth process and a pure jump process with finitely many jumps. Formulas for the mean number of both continuous and discontinuous crossings through a fixed level on a compact time interval are obtained. We present examples in which we compute explicitly the mean number of crossings and compare which kind of crossings dominates for high levels. In one of the examples the leading term of the tail of the distribution function of the maximum of the process over a compact time interval as the level goes to infinity is obtained. We end giving a generalization, to the non-stationary case, of Borovkov-Last’s Rice Formula for Piecewise Deterministic Markov Processes.
KeywordsRice formula Level crossings Process with jumps Number of continuous and discontinuous crossings
AMS 2000 Subject ClassificationsPrimary—60G10 stationary processes 60J75 Jump processes Secondary—60G70 Extreme value theory; extremal processes
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